Prelim%202%20Solutions - Economics 3010 Fall 2008 Professor...

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Unformatted text preview: Economics 3010 Fall 2008 Professor Daniel Benjamin Cornell University EXAM #2 Credit Guide I. True / False / Uncertain Your grade on these questions will depend on the generality, completeness, and persuasiveness of your explanation, not simply on whether the “true” or “false” is correct. The objective here is to provide an answer that convinces; not merely an answer that is “not wrong.” At the very least, make sure that you give clear definitions for the relevant economic terms used in the question. Please use a separate blue book for this section. (5 points) 1) Moral codes and charities are among the private solutions to the externality problem. True. 2 points: An externality arises when a person engages in an activity that influences the well‐being of a bystander and yet neither pays nor receives any compensation for that effect. 1 point: The problem when there is a positive externality is that private markets underprovide the good or service / when there is a negative externality, overprovide. 1 point: An example of a moral code helping to solve the problem. E.g., A moral code of not littering can help reduce the negative externality from littering. 1 point: An example of a charity helping to solve the problem. E.g., the Sierra Club (which promotes environmental causes) is largely funded by private, charitable contributions. (5 points) 2) If the production function is smooth and the solution to the firm’s cost‐ minimization is interior (i.e., not a corner solution), the technical rate of substitution is equal to the factor price ratio. True. 2 points: The technical rate of substitution (TRS) is the slope of the isoquant. 3 points: Derivation of the firm’s profit‐maximization condition and/or graphical illustration showing tangency between the isoquant and the isocost line. (5 points) 3) In a perfectly competitive industry, every firm earns an economic profit of zero in both the short run and the long run. Uncertain. 2 points: In the short run, the equilibrium price may differ from average cost, in which case a perfectly competitive firm may earn profits or losses. 1 2 points: In the long run, if there is free entry or exit, the number of firms will adjust until economic profit equals zero. 1 point: In some cases, there may not be zero profit even in the long run. For example, if there are restrictions on entry (like licensing of taxicabs) and if the firm did not have to pay for the license, then the firm may earn positive economic profit. (5 points) 4) A firm with production function f(K, L) = K0.9 L0.8 has increasing returns to scale and decreasing marginal product of labor. True. 1 point: Increasing returns to scale means that multiplying all inputs by a factor of λ > 1 increases output by a factor greater than λ. 1.5 points: This production function has increasing returns to scale: f(λK, λL) = (λK)0.9 (λL)0.8 = λ1.7 K0.9 L0.8 = λ1.7 f(K, L) > λ f(K, L). (Alternatively, a Cobb‐Douglas production function, f(K, L) = Ka Lb, has increasing returns to scale if and only if a + b > 1, which is true here.) 1 point: Decreasing marginal product of labor means that MPL = ∂f(K, L) / ∂L is decreasing in L. 1.5 points: This production function has decreasing marginal product of labor: ∂f(K, L) / ∂L = 0.8 K0.9 / L0.2, which is smaller when L is larger. (Alternatively, a Cobb‐Douglas production function, f(K, L) = Ka Lb, has decreasing marginal product of labor if and only if b < 1, which is true here.) (5 points) 5) According to the Second Fundamental Theorem of Welfare Economics, any Walrasian equilibrium is Pareto efficient. False. 1 point: It is the First Fundamental Theorem of Welfare Economics that says that any Walrasian equilibrium is Pareto efficient. 2 points: The Second Fundamental Theorem of Welfare Economics says that under certain conditions, every Pareto efficient allocation can be achieved as a Walrasian equilibrium by an appropriate lump‐sum redistribution of the initial endowment. 2 points: (Further intuition or an appropriate graph.) II. Brief Problem (5 points) Every year for as long as anyone can remember, Mr. Wu has decorated his lawn every holiday season with a spectacular winter scene. Tourists would come from far and wide just to see, admire, and photograph his lawn show. This year, Mr. Wu has decided to retire his lawn show because, even though he personally gets $500 worth of enjoyment from it, it now costs him $600 worth of effort to put the scene together (since he isn’t getting any younger). If there were one “large” tourist – say a local TV station – that received $200 of benefit from the show and could bargain with Mr. Wu at low cost, what would the Coase theorem predict? (To receive full credit, make sure you state the Coase theorem.) Why might the Coase theorem not apply if, instead of one “large” 2 tourist, there were 100 “small” tourists – say families – each of whom received $20 of benefit from the lawn show? 1 point: The Coase theorem states that if private parties can bargain over the allocation of resources and if there are no transactions costs, then bargaining will lead to an efficient outcome, regardless of the initial allocation of property rights. 2 points: If there were one “large” tourist that received $200 of benefit from the show, the Coase theorem predicts that the show will be provided because the tourist will pay, say, $150 to Mr. Wu. (Any amount between $100 and $200 will work.) 2 points: If instead there were 100 “small” tourists, the Coase theorem may not apply because the transactions costs of organizing such a large group of people to negotiate with Mr. Wu may be too large. III. Multi‐Part Problem Note: This question is designed so that if you skip part of the question, you still have enough information (stated explicitly earlier in the question) to answer later parts of the question. Please answer as many parts as you can. Suppose the market for New York City taxicabs is perfectly competitive. Each firm requires a car to use as a taxi (capital K), but then the amount of taxi services provided depends on the number of hours worked as a driver (labor L). Specifically, each firm in the industry has production function: if K = 1 q(K, L) = { 2 L1/2 { 0 if K = 0, where q is taxi services per day, K is the presence or absence of a car, and L is number of hours worked as a driver. The rental rate of a car is $100 per day, and the wage rate is $1. Daily market demand for taxi services is D(p) = 2,000 – 10p, where p is the price. It takes time to sign a rental contract on a car, and firms that are currently renting cars are stuck in annual contracts that cannot be cancelled on short notice. In the short run, there are 75 firms that are currently renting cars. (We will assume for simplicity that each firm can have only zero cars or one car, but not more than one car.) (5 points) (a) Show that a firm in this industry has the following short run cost function: SRTC(q) = 100 + (q2 / 4) for any q ≥ 0. 1 point: The cost function gives the minimal cost for producing a given level of output. (Full credit for either stating this or demonstrating understanding of it from the calculations that come next.) 2 points: The “100” part of the SRTC is the (fixed) cost of renting a car. 2 points: Producing q units of output requires enough labor to satisfy q = 2 L1/2, which is L = q2 / 4. Since the wage rate is $1, this adds the “q2 / 4” (the variable cost) of SRTC. (6 points) (b) What is the firm’s short run average cost (SRAC) function? Short run average variable cost (SRAVC) function? Show that the firm’s short run marginal cost function is 3 SRMC(q) = q / 2 for any q ≥ 0. 2 points: The firms’ short run average cost (SRAC) function is SRAC(q) = SRTC(q) / q = (100 / q) + (q / 4). 2 points: The firms’ short run average variable cost (SRAVC) function is the variable cost part of SRAC(q), SRAVC(q) = (q / 4). 2 points: The firms’ short run marginal cost (SRMC) function is SRMC(q) = ∂SRTC(q) / ∂q = q / 2. (5 points) (c) Graph the firm’s SRAC, SRAVC, and SRMC curves. With reference to your graph, explain why no firm will ever shut down in the short run, regardless of the output price. 3 points: [accurate diagram] 1 point: A firm will shut down when SRAVC(q) > p at the profit‐maximizing q. 1 point: But since the SRMC curve lies above the SRAVC curve at all output levels, the firm’s shut‐down condition will never apply. (3 points) (d) Show that each firm’s short run supply function is Sfirm(p) = 2p and that the short‐ run market supply function is Smkt(p) = 150p. 1 point: The firm’s short run supply function is its SRMC curve. 1 point: Since SRMC(q) = q / 2, Sfirm(p) = 2p. 1 point: Since there are 75 identical firms, Smkt(p) = 75 × Sfirm(p) = 150p. (3 points) (e) Show that the short run equilibrium market price is pSR = 12.5 and that each firm produces qSR = 25. At the short run equilibrium, what is a representative firm’s profit? 1 point: Setting Smkt(p) = 150p = 2,000 – 10p = D(p) gives pSR = 12.5. 1 point: Hence qSR = Smkt(12.5) = D(12.5) = 25. 1 point: A representative firm’s profit is pSR qSR – SRTC(qSR) = (12.5)(25) – (100 + 252/4) = 56.25. (No points off for miscalculation, as long as clearly understood conceptually.) (5 points) (f) In the long run, new firms can rent cars, and old firms can get out of their rental contracts. Explain why the long run cost function is LRTC(q) = { 100 + (q2 / 4) if q > 0 { 0 if q = 0. Show that the long run equilibrium price is pLR = 10 and that each firm produces qLR = 20. 1 point: In the long run, a firm can produce zero output at zero cost (by getting out of its rental contract). Otherwise, the costs are the same as in the short run. (The cost of renting a car is a quasi‐ fixed cost that must be paid to operate, even in the long run.) 1 point: The zero profit condition implies that the firm’s long run equilibrium output is the level that minimizes long run average cost. 1 point: Long run average cost, LRAC(q), equals (100 / q) + (q / 4). 4 1 point: Setting ∂LRAC(q) / ∂q = ‐(100 / q2) + (1 / 4) = 0 implies that qLR = 20. 1 point: The long run equilibrium price is pLR = LRAC(qLR) = 10. (This price ensures that firms earn zero profit.) (4 points) (g) Draw a supply and demand diagram depicting the long run equilibrium, and explain why the long run supply curve is a horizontal line (i.e., perfectly elastic) at the long run equilibrium price. Show that the long run equilibrium market quantity of taxi services provided is QLR = 1900. What is the long run equilibrium number of firms? 1 point: [accurate diagram] 1 point: The long run supply curve is horizontal because the number of firms will adjust to make sure that firms are earning zero profit, which requires that price equals average. Hence the price is fixed in the long run. 1 point: The long run equilibrium market quantity of taxi services provided is D(pLR) = 2,000 – 10(10) = 1900. 1 point: The long run equilibrium number of firms is the market quantity of taxi services divided by the quantity per firm: 1900 / 20 = 95. (No points off for miscalculation, as long as clearly understood conceptually.) (4 points) (h) Suppose that production of each unit of taxi services causes $5 of pollution damage (due to car exhaust) so that the marginal social cost of production is $5 higher than the marginal private cost at every level of output. Draw a supply and demand diagram depicting the demand curve (i.e., the marginal benefit curve), the long run marginal private cost curve, and the long run marginal social cost curve. Show that the socially optimal market quantity of taxi services provided is Qsocial = 1850. 2 points: [accurate diagram] 2 points: The long run average social cost of production (taking into account the externality) is $10 + $5 = $15. Hence the socially optimal market quantity of taxi services provided is Qsocial = D(15) = 2,000 – 10(15) = 1850. 5 ...
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